- #1

spaghetti3451

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I have literally no idea how to begin! It's just that with these theoretical problems there's no straightforward starting point!

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In summary, the conversation discusses the difficulty of proving the uniqueness of components of a vector with respect to a given basis. One approach is to assume they are not unique and then use the definition of independence to show that the coefficients must be zero, ultimately proving uniqueness. The conversation also mentions the challenge of finding a starting point for such theoretical problems and the general pattern of proving uniqueness by showing two elements are equal.

- #1

spaghetti3451

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I have literally no idea how to begin! It's just that with these theoretical problems there's no straightforward starting point!

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- #2

HallsofIvy

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[tex]v= a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_nv_n[/tex]

and

[tex]v= b_1v_1+ b_2v_2+ \cdot\cdot\cdot+ b_nv_n[/tex]

Now subtract:

[tex](a_1- b_1)v_1+ (a_2- b_2)v_2+ \cdot\cdot\cdot+ (a_n- b_n)v_n= 0[/tex]

One of the requirements for a "basis" is that the vectors are independent. Use the definition of "independent".

- #3

spaghetti3451

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I have to tell you, though! It's not easy to figure out how to go about the problem.

I just wish the solution were as obvious as it looks now!

- #4

dodo

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- #5

blue_raver22

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I understand the frustration and difficulty in proving theoretical concepts. In order to prove the uniqueness of vector components with respect to a given basis, it is important to first define and understand the concept of a basis. A basis is a set of linearly independent vectors that span a vector space. This means that any vector in the vector space can be expressed as a linear combination of the basis vectors.

To prove the uniqueness of vector components, we can start by assuming that there are two sets of components for the same vector with respect to a given basis. Let's call these two sets of components A and B. Now, we can express the vector using both sets of components as:

v = A1b1 + A2b2 + ... + Anbn = B1b1 + B2b2 + ... + Bnbn

where A1, A2, ..., An and B1, B2, ..., Bn are the components of the vector with respect to the basis vectors b1, b2, ..., bn.

Since the basis vectors are linearly independent, we can equate the coefficients of each basis vector on both sides of the equation. This will give us n equations with n unknowns, which can be solved to show that A1 = B1, A2 = B2, ..., An = Bn. This proves that the components of a vector with respect to a given basis are unique.

In addition, we can also use the concept of linear independence to show that any other set of components for the same vector with respect to the given basis will also be the same as A and B. This further strengthens the uniqueness of vector components.

In conclusion, proving the uniqueness of vector components with respect to a given basis requires a deep understanding of the concept of a basis and the use of linear independence. It may also require the use of mathematical techniques such as solving equations to show the uniqueness of the components. I hope this explanation helps in tackling this difficult theoretical problem.

Basis vectors are a set of linearly independent vectors that are used to describe a vector space. They are used to form a coordinate system and are typically chosen to be orthogonal or at right angles to each other.

Solving problems on basis vectors can be difficult because it requires a deep understanding of linear algebra and vector spaces. These problems can involve complex mathematical equations and may require the use of advanced techniques such as eigenvalues and eigenvectors.

In physics and engineering, basis vectors are used to describe the orientation and direction of physical quantities such as force, velocity, and acceleration. They are also used in fields such as computer graphics and robotics to represent transformations and rotations in space.

One example of a difficult theoretical problem on basis vectors is finding the eigenvalues and eigenvectors of a large matrix. This problem is important in many applications such as data analysis and optimization, but can be challenging due to the size and complexity of the matrix.

To improve your understanding of basis vectors, it is important to have a strong foundation in linear algebra and vector spaces. You can also practice solving problems and working with different types of basis vectors in various contexts. Additionally, seeking guidance from a mentor or attending workshops and seminars can also help deepen your understanding of this topic.

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